- #1
Izzhov
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Homework Statement
A particle is sliding inside a frictionless paraboloid defined by [itex]r^2 = az[/itex] with no gravity. We must show that the force of constraint is proportional to [itex](1+4r^2/a^2)^{-3/2}[/itex]
Homework Equations
[itex]f(r,z) = r^2-az = 0[/itex]
[itex]F_r = \lambda \frac{\partial f}{\partial r}[/itex] (and similarly for [itex]F_z[/itex])
The Attempt at a Solution
[itex]F_r = \lambda \frac{\partial f}{\partial r} = 2r\lambda[/itex] and [itex]F_z = \lambda \frac{\partial f}{\partial z} = -a\lambda[/itex] and hence the total force of constraint is [itex]F = \sqrt{F_r^2+F_z^2} = (4r^2+a^2)^{1/2}\lambda[/itex].
So you can factor out the a from the square root and get [itex](1+4r^2/a^2)^{1/2} a \lambda[/itex] but the exponent is supposed to be -3/2, not 1/2. What am I missing? Am I supposed to find a factor of [itex](1+4r^2/a^2)^{-2}[/itex] in the [itex]\lambda[/itex] somewhere? How would I go about doing that?